Scientific Machine Learning (SciML) has emerged as a powerful approach for Model Order Reduction (MOR) and operator learning in physical systems with complex temporal evolution and parametric variability [1,2]. In this work, we introduce Conditioned Message Passing (CoMPass) [3], a unified framework connecting graph-based, meshless, and operator-learning methods exploiting the conditioning paradigm with respect to problem data, external fields, or latent dynamical states. We formally show that CoMPass generalizes several existing architectures, including Latent Dynamics Networks (LDNets) and Neural Implicit Flows (NIFs), enabling their extension to operator learning tasks. Within this framework, we introduce FiLM-Graph Neural Operators (FiLM-GNOs), a parameter-efficient class of neural operators based on feature-wise linear modulation, inspired by classical separation-of-variables techniques. The framework bridges mesh-based and mesh-free MOR approaches, interpreting meshless methods as neural operators with a degenerate quadrature approximations of their kernel. We validate CoMPass on challenging MOR and operator learning benchmarks in computational mechanics, including time-dependent systems with varying geometries and complex parameterizations. The proposed framework achieves state-of-the-art performance while requiring up to two orders of magnitude fewer trainable parameters. These results establish CoMPass as a flexible, expressive and theoretically grounded framework for SciML, providing a scalable foundation for future developments in data-driven physical modeling. [1] Berner J., Liu-Schiaffini M., Kossaifi J., Duruisseaux V., Bonev B., Azizzadenesheli K., Anandkumar A., "Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning", arXiv:2506.10973 [cs], 2025. [2] Regazzoni F., Pagani S., Salvador M., Dede’ L., Quarteroni A., "Learning the intrinsic dynamics of spatio-temporal processes through Latent Dynamics Networks", Nature Communications, Vol. 15, p. 1834, 2024. [3] Tomada L., Pichi F., Rozza G., "Conditioned Message Passing for Model Order Reduction and Operator Learning, In preparation".